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Economics Letters North-Holland 35 (1991) 273-277 273 Unit roots and seasonal unit roots in macroeconomic time series Canadian evidence Hahn Shik Lee Bureau of the Census, Washington, DC 20233, USA Pierre L. Siklos Wilfrid Laker Received Accepted Uniuersity, 15 August 2 October Waferloo, Ont. N2L 3C5, Canada 1990 1990 Unit root tests are conducted to determine whether the unit root found in seasonally adjusted data may be due to seasonal adjustment filters typically apptied by government agencies (i.e., ARIMA X-11). We find that a unit root exists in both the raw and adjusted data (except for the unemployment rate). Second, we also find that the commonly used seasonal adjustment filter introduced by Box and Jenkins (1970) is generally inappropriate. 1. Introduction The vast literature on unit roots attests to the somewhat strong belief that shocks to many aggregate times series tend to be permanent in nature, although any such consensus has shown signs of weakening [e.g., see Cochrane (1988) and Perron (1989) but see also Kormendi and Meguire (1990)]. Part of the interest in testing for unit roots stems from the work of Box and Jenkins (1970) who suggested that many time series are difference stationary. While the focus of recent research has been about the time series properties of seasonally adjusted time series, the focus of attention has lately reverted back to the implications of using seasonally adjusted versus unadjusted data after some neglect over the past few years. For example, Barsky and Miron (1989) suggest that the routine elimination of the seasonal component of time series by well known methods, such as ARIMA X-11 [see Hylleberg (1986, ch. 5) for a survey], results in the omission of an important source of macroeconomic fluctuations. Furthermore, since economists prefer testing their theories using data expressed in real terms, it is not apparently the case that the common practice of using seasonally adjusted time series, which are then deflated by the appropriate price index, has non-neutral consequences on estimates from econometric models [Ghysels and Karangwa (1990)]. Moreover, the unit root commonly found in postwar U.S. GNP may, in part, arise from the use of seasonally adjusted data [Ghysels (1990)]. The problem with U.S. data, in particular, is the relative 0165-1765/91/$03.50 0 1991 - Elsevier Science Publishers B.V. (North-Holland) H.S. Lee, P.L. S~klos / Unit roots and seasonal unit roots 274 difficulty in obtaining high quality data in both seasonally adjusted and unadjusted forms [e.g., for U.S. GNP see Ghysels (1990, Appendix A) and Barsky and Miron (1989, Appendix A). Finally, there is considerable interest in whether aggregate time series contain a unit root at some of the seasonal frequencies, as well as a unit root at the long-run (or zero) frequency found in seasonally adjusted data. The present study considers several postwar quarterly Canadian macroeconomic series, which are widely available both in seasonally adjusted and unadjusted forms to examine the sensitivity of the unit root result in seasonally adjusted data to seasonal adjustment procedures (i.e., ARIMA X-11) applied by government agencies. Some of our results can be summarized as follows. We first find that the unit root at the zero frequency found in seasonally unadjusted data is also present in seasonally adjusted data, confirming that seasonal adjustment via the ARIMA X-11 does not eliminate the unit roots at the zero frequency found in the original time series data. We also find evidence for the presence of roots at some seasonal frequencies in seasonally unadjusted data, while the same is not true for seasonally adjusted data, indicating that the X-11 method does remove seasonal unit roots, if any, in the raw data. The presence of unit roots at seasonal frequencies found in seasonally unadjusted data, however, turns out to be sensitive to the removal of deterministic seasonality. Also, there are some differences in the test results between real and nominal magnitudes. For example, seasonally unadjusted real per capita GDP possibly has a unit root at the biannual frequency, while nominal per capita GDP does not. We also find that most series, with the exception of the narrow money stock (Ml) and GDP, do not contain a seasonal unit root at the annual frequency. 2. Testing procedure Tests and issues surrounding the existence of unit roots in seasonally adjusted data are so well known that there is little need to dwell on the main points. A common test procedure is to generate the so-called augmented Dickey-Fuller (DF) test by estimating the following regression: P Ax, = (Y+ ,f?t + 7x,-i + c a,Ax,_, + ~1, j=l (1) where x is a time series, often expressed in the logarithm of the levels, t is a time trend, and A is the difference operator (i.e., Ax, - x,-i). Under the null hypothesis, series x is postulated to have a unit root with no drift, acceptance of which requires that the t-statistic on the coefficient 7 not be statistically different from zero (as well as /? if a trend is included). Critical values for this ‘t’ test may be found in Fuller (1976, Table 8.5.2.). Since the order of the autoregression is unknown it was chosen on the basis of the well known Akaike and Schwarz information criteria. Drawbacks with the DF test include the possible presence of a moving average term in E, in eq. (l), and the failure of the autoregressive correction to increase with sample size [see Schwert (1987)]. Accordingly, the unit root test suggested by Stock and Watson (1986) was also implemented. For the seasonally unadjusted series, the procedure developed by Hylleberg, Engle, Granger and Yoo (1990; hereafter HEGY) was used, which consists in estimating the following regression: p is a constant, t a time trend, ‘k(L) is a polynominal autoregression correction terms as in eq. (1) difference D, are the deterministic seasonal dummies and A4x, = x, - x,_~, that is, the fourth-order 215 H.S. Lee, P. L. Siklos / Unit roots and seasonal unit roots of series x,. The terms Y,,, to Y,,, reflect decomposed as follows, namely: (1 - L)(l the fact that + L + L2 + L3)x, = (1 - L)Y,,,, a fourth-order difference may be further (3) (1-L)(1-L+L2-L3)X,=(1+L)Y2,r, (4) (1 + L2)(1 - LZ)x, = (1 + L*)Y3,r. (5) Eqs. (3) to (5) define the existence of four possible unit roots in seasonally unadjusted data. Eq. (3) represents that part of x, free of seasonal unit roots, while expressions (4) and (5) suggests that a unit root can exist at the biannual and annual frequencies, respectively. Therefore, in estimating eq. (2) the unit root found in seasonally adjusted data signifies accepting the null that ZZ, is zero. A finding that II2 is zero implies the existence of a unit root at the twice yearly seasonal cycle, while, if both II3 and II, are zero, there is a unit root at the annual seasonal cycle. HEGY (1990) give critical Miron (1990) have recently values for the foregoing test based on quarterly data, while Beaulieu-and extended the procedure to monthly data. 3. Empirical results Given the large volume of test results only a summary is provided in table 1. The detailed statistics are available upon request. All the data are from CANSIM (Canadian Socio-Economic Information Management System), and are also available upon request. The table indicates for the series considered whether they are integrated of order one, that is Z(l), at the zero and seasonal frequencies. The results for seasonally unadjusted data may be influenced by whether or not some seasonality is deterministic in nature. Accordingly, eq. (2) was estimated with and without seasonal dummies. If seasonality indeed arises regularly, it may be better captured by the practice of adding deterministic seasonal dummies, when no unit roots appear at the seasonal frequencies. The unit root tests for all the series listed in table 1 were applied to data in the logarithm of the levels, except for the unemployment rate and interest rate series which, following previous convention, are in levels. Whenever inference was sensitive to either the type of unit root test conducted, or to the lag length of the autoregressive correction factors in eqs. (1) and (2) an asterisk appears next to the outcome of the test. Finally, all inference is based on the 5% significance level. A comparison of columns (l), (2), and (5) in table 1 reveals that the unit root finding in seasonally adjusted data is not generally affected by seasonal adjustment at the source. Thus, a series x, remains Z(1) at the zero frequency whether the data are in raw or adjusted forms. One notable exception, however, is the unemployment rate which is Z(1) at the zero frequency in seasonally adjusted data but is Z(0) for unadjusted data. By contrast, and unlike Ghysels findings based on US data, ’ GDP in its various representations is consistently Z(1). The apparent difference in the behaviour of GDP versus the unemployment rate is interesting because it suggests that while shocks to GDP are permanent, shocks to the unemployment rate may be transitory. Blanchard and Quah (1989) have recently built and tested a model with such features by supposing that aggregate supply shocks are transmitted through the unemployment rate while aggregate demand side shocks influence output. ’ Ghysels does not, however, use the HEGY test to arrive at his conclusions. For the unemployment series, one would conclude the series to be I(1) in the case where no seasonal dummies are included and when the autoregressive correction term is not of order one. 276 H.S. Lee, P. L. Siklos / Unii roots and seasonal unit roots Table 1 Unit roots and seasonal unit roots in Canadian Series frequency: Seasonally adjusted (1) Nominal GDP I(I) Real GDP I(I) Per cap nom GDP I(1) Real per cap GDP I(1) Implicit price deflator I(1) Imports I(1) Real imports I(1) Exports I(1) Real exports I(1) Money supply (Ml) I(1) Real Ml I(l) Unemployment rate I(1) Stock prices I(1) Treasury bill rate (TB) I(1) Ex post real TB I(0) Long term bond rate (LTBRJ(1) Ex post real LTBR I(0) Exchange rate I(I) Real exchange rate I(I) macroeconomic time series. a Seasonally unadjusted no seasonal dummies Seasonally unadjusted seasonal dummies Zero Biannual Annual Zero Biannual Annual (2) (3) (4) (5) (6) (7) I(1) I(I) I(1) I(1) I(1) l(I) I(0) b I(1) I(1) I(1) I(1) I(0) h I(1) I(1) I(0) I(1) I(0) I(I) I(1) I(O) I(l) I(0) I(1) I(0) I(1) I(I) I(1) I(0) I(1) I(1) Ill) I(O) I(0) I(0) I(0) I(0) I(0) I(0) b I(I) I(1) I(1) I(1) I(0) I(0) I(O) I(O) I(0) I(1) I(0) I(0) I(0) I(0) I(0) I(0) I(0) I(0) I(0) I(l) I(I) I(1) I(I) I(1) I(1) I(O) I(I) I(I) I(1) I(1) I(0) I(I) I(I) I(0) I(1) I(0) I(I) I(I) I(0) h I(1) I(0) I(1) I(0) I(0) I(0) I(0) I(I) I(0) I(0) I(0) I(0) I(0) I(0) I(0) I(0) I(O) I(I) b I(0) b I(I) I(I) l(I) I(0) I(0) I(0) I(0) I(0) I(0) l(0) l(0) l(0) I(0) I(0) l(0) l(O) I(0) I(0) Sample is 1947 (I)-1984 (IV), except for the money supply unadjusted; 1966 (I)-1987 (Iv) adjusted], the exchange Prices [Standard and Poor’s 500 Index, 1956 (I)-1987 Government of Canada bonds, 1948 (Q-1987 (IV)]. In all in logarithms of the levels, except for the unemployment levels, Signifies that inference is sensitive to either the lag length type of unit root test conducted. b b b b b [1953 (I)-1987 (IV)], the unemployment rate [1962 (I)-1987 (IV), rate [closing spot versus U.S. dollar, 1970 (ll)-1987 (Iv)], Stock (Iv), and the long-term bond rate (yield on 10 years and over cases inference is based on a 5% level of significance. All series are rate, the Treasury bill rate, the long-term bond rate, which are in specification, based on the Akaike and Schwarz criteria, or to the A glance down columns (4) and (7) suggests that, regardless whether one believes that some seasonality is deterministic in nature, there are relatively few macroeconomic time series which appear to process a unit root at the annual seasonal cycle. Hence, the A4x, filter recommended by Box and Jenkins (1970) appears not to be warranted. A few more series display a unit root at the biannual cycle (column (3)) but this result is, as expected, sensitive to the removal of deterministic seasonality (column (6)). Note also that, as one would expect, interest rates and the nominal exchange rate possess no seasonal unit roots, while ex post real interest rates 2 are Z(0). Finally, there are some differences, at the seasonal frequencies, between time series expressed in nominal and real terms. For example, nominal and real GDP, either in log levels or in per capita forms, are Z(0) and Z(l), respectively, at the biannual seasonal cycle. Thus, using nominal series which are then routinely deflated may have consequences in inferences about economic relationships [see also Ghysels and Karangwa (1989)]. * Computed as the nominal interest rate level less the log first difference in the consumer price index H.S. Lee, P.L. Sikh / Unit roots and seasonal unit roots 277 4. Conclusions The use of seasonally unadjusted data, with one possible exception, does not influence the finding of a unit root in seasonally adjusted data. However, there are few series which exhibit a seasonal unit root at the annual cycle thereby casting doubts on the appropriateness of a commonly used data filter for unadjusted data advocated by Box and Jenkins (1970). Nevertheless, in some instances, most notably GDP and the money supply (Ml), a seasonal unit root is present, suggesting perhaps the possibility that some seasonal unit roots may be common. In other words, some series may be cointegrated at some of the seasonal frequencies. This line of research was initiated by Engle, Granger, and Hallman (1989) and HEGY (1990) and is currently being pursued by, among others, Lee (1989) Lee and Siklos (1990), and Ghysels, Lee, and Siklos (1990). References Barsky, R.B., and J.A. Miron, 1989, The seasonal cycle and the business cycle, Journal of Political Economy 97, 503-534. 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Siklos, 1990, A survey of seasonality, seasonal unit roots, and seasonal cointegration in Canadian and U.S. Data, in progress. Ghysels, E., 1990, Unit root tests and the statistical pitfalls of seasonal adjustment: The case of U.S. postwar real gross national product, Journal of Business and Economic Statistics 8, April, 145-152. Ghysels, E. and E. Karangwa, 1989, ‘Nominal’ versus ‘real’ seasonal adjustment, Cahier 2188 (Dtpartement des sciences economiques et centre de recherche et developpement en tconomique, Universitt de Montreal, Montreal, Ont.). Hylleberg, S., R.F. Engle, C.W.J. Granger and B.S. Yoo, 1990, Seasonal integration and cointegration, Journal of Econometrics, forthcoming. Hylleberg, S., 1986, Seasonahty in Regression (Academic Press, Orlando, FL). Kormendi, R.C. and P. Meguire, 1990, A multicountry characterization of the nonstationarity of aggregate output, Journal of Money, Credit and Banking 22, 77-93. Lee, H.S., 1989, Maximum likelihood inference on cointegration, Unpublished manuscript (University of California, San Diego, CA). Lee, H.S. and P.L. Siklos, 1990, The influence of seasonal adjustment on unit roots and cointegration: The case of the consumption function for Canada, 1947-1987, Unpublished manuscript. Perron, P. 1989, The Great Crash, the Oil Price Shock and the unit root hypothesis, Econometrica 57, 1361-1402. Schwert, G.W., 1987, Effects of model specification tests for unit Roots in Macroeconomic data, Journal of Monetary Economics 20, 73-103. Stock, J.H. and M.W. Watson, 1986, Does GNP have a unit root? Economics Letters 22, 147-151.